the radical rational…

in search of innovative ideas with a well-balanced approach for the math classroom

A Struggle with Function Inverses

So as we were doing this today – simple enough using inverse operations to find the inverse function…

But as we did this one f(x) = (x-5)^2 to this f^-1(x) = sqrt(x) + 5

Student question… because you had x-5 in parenthesis, won’t the +5 also be inside the radical?

I’m just curious how anyone else responds to this.

We picked #’s and evaluated the expression, modeling transformation on a number line…but referring to inverses as a way to return where we started.  

f(8)=(8-5)^2 = 9

Simple enough…working backwards… sqrt(9)=3+5=8…back where we started x=8.

But what happens if x=3…


Working backwards…sqrt(4) = 2+5 = 7…not where we started at x=3.

Is this where you talk about restricting the domain in order for the inverse function to be defined?

Please don’t judge, I’ve taught straight up procedures in the past, even focusing on mostly linear inverses.  But I want my student to understand what they’re doing and why and be able to expand their understanding to more complex scenarios on their own.

Listening to them discuss this initial question, made me realize how Building Functions and Transformations has huge impact on foundational understanding.

The other realization today…very few of my students are comfortable and fluent in equivalent expressions. 

For example…
f(x)= 6-2x some asked if they could rewrite -2x+6? Sure!

But then when discussing their results, we saw several equivalents…but they did not recognize, rather argued others were incorrect.


We picked a value for x, then tested to see if they were equivalent.  A few were a bit perplexed they were the same value. 

This is one of those assumptions I’ve made but realize we need to address/refresh.

Would a matching, always/sometimes/never be sufficient?


Posting Learning Targets yay or nay

Thanks to @JustinAion,  I got thinking…

It depends… on my class and the students and the activity…to determine if I actually post it.

However, when I do, I refer to it at the beginning, throughout the task – to remind students of the end goal, and again as a wrap up – whether reflection, exit ticket of discussion/summary to end class.  And I like to refer to it the following day as we begin the next lesson, just as a quick review.

I, personally, would prefer to have an overarching Essential Question for each lesson to use rather than a specifically stated target.  However, I sometimes struggle a lot with Writing EQs, would love a colleague to collaborate on these.

Here’s a section of the unit organizers I’ve used this past year (thanks @lisabej_manitou). 


And a link to this file.
Unit Organizer
Functions Overview

I give them to students toward beginning of unit, we complete the words worth knowing for vocabulary (thanks @mathequalslove). Then read through actual targets.  When quizzes are given back or practice problems checked, students have a place to reflect/record thwir level of learning as well.  Because students have this in their INBs, I can quickly refer to them if not posted on the board on any given day. 

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Inequalities in 2 Triangles

After some discussion about the hingle theorem… the question was …since 40° is twice the size of 20°, won’t side e be twice as long as side d?


Well, we found some counterexamples to prove it wouldnt always be true.

But what about if the triangles are Isosceles?  The picture is actually marked wrong…all of those sides should be marked congruent. 

Two students said it would always work…they believe if  the triangles are isosceles, the third side will be proportional to the angles. 
My projector bulb is shot.  So I could not pull up geogebra to play around in class.  I told them we’d explore tonight and begin there tomorrow.


Trash the Lesson

Literally. We trashed a lesson today.  It was a lesson I’ve used for a long, long time.  Triangle Midsegment Investigation.  In recent years, I’ve used geogebra to test/ explore student findings.

It lead a small group of students through realizing midsegment is parallel and 1/2 the length of the corresponding base.


For whatever reason, it was not sinking in. 

I was getting frustrated.  On the edge of sarcasm.  I remembered a comment from my colleague last Friday, to remember to have fun with them.

The lesson wasn’t clicking with them. I wanted to trash it.

So, I picked up a student’s paper, wadded it up and instructed the entire class to follow my lead.

Noone was allowed to leave class until they threw a paper wad at someone.

They laughed. I LAUGHED.  It was totally worth it.

Then I grabbed some poster graph paper and they opened their INBs…we got the big ideas and went on to use them in a few problems…successfully.

A couple of times students made comments how all we had been doing was coming together in the new problems. They were seeing the connections.

Yep. Sometimes we just need to trash the lesson and move on.

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Painting a Bridge

In my Algebra I, we are looking at parent functions. Students said this week was quite easy, they felt they were doing 3rd grade work.  But I assured them
recognizing the parent equation and making connections to the parent graphs may seem easy, but it’s a lead-in to more intense math!

We’ve done several data collections throughout the semester, mostly linear, a few quadratic and exponential.   But today we took a look at rational with Painting the Bridge, which is embedded in a MARS lesson. 

Students are asked to sketch the relationship x:# workers and y: # hours each works to complete the given job.






Those are a good overview of what we saw.  I allowed students to ask questions about things they wondered about others’ graphs.  At first glance, a couple of the graphs may look odd, but given the chance to share thwir thimking, student reasoning made perfect sense in the real world.

Though I didn’t have an actual student create this graph, I included it on the board.


I followed the suggested questioning in the MARS lesson, which led most students to some A-ha moments.  What does point Q mean? Points S? Does it make more sense for the graph be solid or dotted? Why?

As a data collection to follow up this discussion, we picked up erasers. One student held a cup in their dominant hand and picked up one eraser at a time and placed it in the cup, we timed.  Then another student helped.  Continued adding workers and it eventually became too crowded, they were dropping erasers and slowed them down.


We compared the shape of our scatter plot and decided maybe exponential or quadrant 1 of a rational (inverse) function.

The calculator power regression resulted in
y =76x^-1.  Which gave us a chance to discuss that -1 exponent.  How it meant the inverse of multiplying by x, which was to divide by x.  So we graphed y=76/x. Nice. They were seeing the connection to our Painting the bridge discussion. 

Oh wait, how many erasers were we picking up? 78. Not bad, huh?

My goal is to give them a concrete data collection for which they can access and connect back to the math.

To end the day, they asked if they could draw a graph on the board and everyone guess the parent function name.  Sure.   They were on task and engaged so I was fine with it.
They began graphing the endpoints of their graphs,  so their classmates were finishing the graph and naming the function. It was humorous. But again, they were engaged.

I love these kids.  They were my favorites today.  It’s been a tough semester at times, but I want to end these last weeks strong. I want them to leave our classroom having grown in confidence and changed their attitude toward math.  That’s my goal. 

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Systems Linear Programming

As an intro to this lesson, I shared this scenario…

You are bidding a contract for Company ABC.  The order is for 12,000 dozen of a product and needs to be completed within 3 months.

First student question, why would anyone need 12,000 dozen of anything?  They felt this amount was a ludicrous number.  (After many summers working at Fruit of the Loom, I knew this was within reason, but a nice discussion anyhow.)

Well, is it?
According to Apple Press Info, if it’s as popular as  iPhone6…no. 

First Weekend iPhone Sales Top 10 Million, Set New Record

We figured if we had equal distribution among all 50 states, this was quite doable.

Do we have the man hours to fulfill this order in 3 months?

There are…21 (bc there were 21 students in class today) workers in this particular unit…who work 8 hours per day, each of you can complete 10 products in 1 hour.  Yes, I just made these up, but that’s what we worked with.

After a few minutes, we started sharing processes, quickly a bit of an argument – why did you do it this way? Should you have….?  Others arrived at the same solution, but with varying approaches.

I could kick myself for not taking a picture of their suggestions.  Some nice verifying one another going on.  However, they were not sure what those values represented…they could get the “right” values but lost when I asked for a label.

Watching students grapple with the numbers, made me realize how far out of reality we’ve taken students math skills.  I just want to do a better job of letting them make sense of problems themselves.

We determined it would cut it close, but we could likely finish this job, maybe requiring a bit of overtime to meet the deadline.

Now, as we make an offer for the contract, what are costs west consider? This leading to an idea of our linear programming. 
Wages, materials, utilities, insurance, packaging,  shipping, etc.  One student even said, there’s a lot to consider. Me, knodding, yes.

Is this a great example intro. Nah. But I feel it’s a nice way to show students there are many options a company must consider prior to the contract, production, sale.

Now, to the hard part.  A variety of students, some with adequate graphing skills, others struggling to find the line x> 3.

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Treats for Teachers (& a November Challenge)

As the day began, members of our student GRIT team began entering classrooms, handing teachers bags of treats.


At first glance, a bag of candy.  But when you opened, you quickly realized it was more than expected.  There were slips of paper included…handwritten notes from students.


The Student GRIT team has copied and dispersed these slips to all 1st period classes, asking each student to tell who and why they were grateful.

What a gift to read short, simple messages from some of your students / former students.

I was so grateful because it’s been a tough year.  Thank you, students for taking the time to do this for us.

They were genuine…and a treasure that has made me realize…I need to tell others thank you.  So, during the month of November, I challenge myself to write one note each day to someone to let them know they’ve made a difference to me.


Battleships & Mines Systems Review


I used Battleships and Mines as a review for systems.  We tested our mine coordinates today in the front lobby.

And then enjoyed the end of the week with the highlighters and blacklights.


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So, my dad is building a storage building.  This is the design he is using with end dimensions for his project.


He asks, “Can your students help me figure out the pitch of the roof? How long should I cut the rafters?  What angles do I use?”

Truth is, my dad likely already has them cut, ready to put together, but he knows there’s a bit of math involved that I might enjoy sharing with my class.

As I read the descritpion, I see this style is a regular octagon. 360° / 8 = 45°.  So the interior angle will be 135°.

When you cut your wood, do I split that angle in half?

I shared the diagram with other geometry teachers and Mr. H, our Carpentry Instructor at LC Area Technology Center.   I wanted to use the right approach/vocabulary Mr. H uses in his courses.  He replied almost immediately. 
The plan is to visit his classroom/shop area soon.

Now, how can I make this real for my students?  Thinking if I give them enough supplies, aka strips of construction paper to model planks of wood, and allow them to create an accurate model, describing processes for finding rafter lengths and angle measures.  Does that make sense?

An opportunity for some use of Trig, or at least a reason to use trig outside the classroom.

The website dad shared has different designs of rafters…which means we have a bank of problems to pull from.

A comment from Mr. H in our back and forth emails…

…one of the hardest to teach…geometry uses what I call the “back angle” measurement (I interpret as interior angle) and carpentry we use the “smaller” angle because we make the cuts at the intersections (if that makes sense)


Mr. H shared some online tools and resources as well to explore.

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Modeling Systems

Sort of a rambling post. But trying to make some sense of my thinking…

I always appreciate posts from @emergentmath.  This particular post made me pause, I had just completed the MARS task, Boomerangs, he references.  We are in the midst of our systems unit.

I used Mary & Alex ‘ s suggestions with beginning systems without the algebra.  Conversations were great, students’ strength in reasoning was evident.

I plan to use Geoff’s suggestion for a matching/sorting activity this werk for students to see the benefits of each type of tool to solve systems.

But where I struggle is with this standard:


I am experiencing some pushback from a handful of students who are able to reason and solve a system without actually modeling it algebraically.

Their reasoning is correct.  They verify their solutions and interpret them correctly.  They can sketch a graph yet “refuse” to model as a system of equations.  I struggle because “their math” is right on.  I realize places where algebraic models can help but I honestly can’t tell them my way is better…yet the standard says…

It feels almost like I am punishing them if I make them model it algebraically.

Then I have others who are not sure where to start.  The equations model provides them a tool, yet they will not embrace it.

How do others handle this situation in your classrooms?

I use graphical, alongside a numerical table of values, with solving/verifying with the equations, letting them see their own connections eventually.

My biggest goal for systems is to provide enough modeling for students to actually “see a context” to connect/make sense of a naked system of equations.

This is where I believe skill/drill has ruined the power and beauty of math.  Finding an intersection point but what in the world does in mean?  It’s a point on a graph. Whoopee.  Why isn’t it all taught in context as a model?

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